Preprints and Publications

We adapt the entropy rigidity result of Besson, Courtois and Gallot (following ideas of Boland and Newberger) to compact quotients of strictly convex real projective manifolds which admit a hyperbolic structure in dimension at least 3. We obtain a similar (though slightly weaker) entropy rigidity statement. Using this and some facts about the Blaschke metric, we prove that the ratio of the Hilbert volume to the hyperbolic volume is uniformly bounded below (by a constant depending only on volume), and that if the projective structure is deformed so that topological entropy of the geodesic flow goes to zero, the volume must go to infinity. We also prove the latter result in dimension 2.

We construct some rank-one manifolds containing twisted 'fat geodesics' -- geodesics which have a uniform flat neighborhood -- but which still contain only a countable collection of closed geodesics. We examine the dynamics of the geodesic flow in such spaces. The paper also contains a closing lemma for 'fat k-flats' which proves that for k>1, having a uniform flat neighborhood of a k-flat implies that there are uncountably many closed k-flats.

We prove a weakened, but still quite useful, version of the specification property for geodesic flows on CAT(-1) spaces. Geodesic flows on compact negatively curved manifolds are an important example of the usual specification property and it can be used to prove many strong results on the dynamics of such flows. With the weak specification property we recover a number of those results in the CAT(-1) setting, including uniqueness of the measure of maximal entropy.

We show that a compact surface equipped with a CAT(κ) metric has Hausdorff dimension 2 and discuss some connections between this regularity results and the dynamics of the geodesic flow. We also discuss entropy rigidity for CAT(-1) manifolds of higher dimension.

We prove some results on ergodic Hilbert transforms of a certain class of functions -- basically mean-zero indicator functions for a finite collection of intervals. We give a connection between everywhere divergence of the transform and Liouville numbers, and construct some Liouville numbers for which the transform converges everywhere.

Proves marked length spectrum rigidity for surfaces with metrics which are nonpositively curved and may have some cone singularities. The angle around each singularity should be >2π. The requirements on one metric may be softened to `non conjugate points' with the addition of an additional hypothesis which may always be true. The proof consists of combining the work of several previous authors on MLS rigidity for surfaces.

We prove marked length spectrum rigidity for geodesic metric spaces with topological dimension 1. These include the easy case of graphs, but also more exotic spaces like Hawaiian earrings and Laakso spaces.

We prove a quantitative shrinking target result for Sturmian sequences derived from circle rotations. You can think of this result as giving some information about the asymptotics of the measure of points whose orbit coding up to step n does not determine the coding at step n+1 (for a special and naturally derived coding). We prove a weak asymptotic and show why a stronger asymptotic fails.

This is an earlier version of the paper above with results for negative curvature only. It is not planned for publication but may be of interest if you only want the negative curvature proof, which is considerably simpler. Only in dimensions 7 & 8 that is it really necessary to prove anything substantial here (see note after Thm 1 in "2-Frame flow...").